A276014 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - z - w)).

1, 72, 45360, 46569600, 59594535000, 86482063571904, 136141986298526208, 226888189910421811200, 394399917777684601926000, 708188604075430924446000000, 1304782547573305224852017990400, 2454776409299366206456800694732800, 4699106882676505497505898579906736000, 9127695522416954472516114289988092800000

Offset: 0

Author

Gheorghe Coserea, Aug 16 2016

Keywords

• nonn

Comments

"The corresponding (order-four) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(4,C)." (see A. Bostan link).

Examples

			1 + 72*x + 45360*x^2 + 46569600*x^3 + ...
		

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..33
• A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (37).
• Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Crossrefs

Cf. A004987, A268549, A268545-A268555.

Programs

Maple

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
end proc:
expr := (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)):
[seq(diag_coeff(expr, i), i=0..14)];

Mathematica

f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y)*(1 - u - v - z - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}] &, f, {x, y, z, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)
Join[{1}, Table[FullSimplify[(4 * 9^n * Gamma[4*n] * Gamma[1/3 + n]) / (Gamma[1/3] * Gamma[n] * Gamma[1 + n]^4)], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 03 2017 *)

Formula

a(n) = [(xyzuvw)^n] (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)).

From Vaclav Kotesovec, Dec 03 2017: (Start)

Recurrence: n^4*a(n) = 24*(2*n - 1)*(3*n - 2)*(4*n - 3)*(4*n - 1)*a(n-1).

For n > 0, a(n) = 4 * 9^n * Gamma(4*n) * Gamma(n + 1/3) / (Gamma(1/3) * Gamma(n) * Gamma(n+1)^4).

a(n) ~ 2^(8*n - 1/2) * 3^(2*n) / (Pi^(3/2) * Gamma(1/3) * n^(13/6)). (End)